Center for Mathematics at Notre Dame
INTERNATIONAL CONFERENCE IN COMMUTATIVE ALGEBRA ANDITS INTERACTION WITH ALGEBRAIC GEOMETRY

Sunday June 16 – Friday June 21, 2019
University of Notre Dame
TALK ABSTRACTS*Talks listed in alphabetical order by speaker name.

Lucho Avramov (University of Nebraska)

Monoids of Betti tables over short Gorenstein algebras
A standard graded algebra $R$ is said to be short if $R_i = 0$ for $i \neq 0,1,2$. In case $R$ is Gorenstein, anexplicit computation of the monoid of Betti tables of $R$-modules will be presented. The result provides new information on
the representation theory of $R$. Some of the techniques developed for its proof apply in more general contexts.
The talk is based on recent joint work with Courtney Gibbons and Roger Wiegand.

Giulio Caviglia (Purdue University)

Upper bounds for Betti numbers in Hilbert schemes of complete intersections.
Let S be a polynomial ring over a field and I a homogeneous ideal containing a regular sequence of known degrees. I will discuss
the existence of sharp upper bounds for the Betti numbers of I in terms of the degrees of the regular sequence and the Hilbert
function, or the Hilbert polynomial, of I. This is a joint work with Alessio Sammartano.

Marc Chardin (Sorbonne Université, France)

Complete intersections in a product of projective spaces
One could define a complete intersection in some irreducible scheme X as a subscheme defined by as many equations as its
codimension. Whenever X is a projective space, this corresponds to subschemes defined by homogeneous complete
intersection ideals; as a consequence many homological invariants are entirely determined by the degrees of these equations. Ina product of projective spaces, the situation is quite different. We will illustrate this on two examples : a hypersurface and thestudy of complete intersection sets of points in a product of two projective spaces. This is based on ongoing joint work with
Navid Nemati.

Aldo Conca (University of Genoa, Italy)

Ideals associated to graphs and special sections of determinantal varieties
According to Lovasz, Saks and Schrijver an orthogonal d-dimensional representation of a graph G is a map from the vertices of G
to the real d-dimensional space sending non-edges to orthogonal vectors. The set of all the orthogonal d-dimensional
representations of G is a variety whose defining ideal, the Lovasz-Saks-Schrijver ideal of G, is generated by the quadrics
expressing the orthogonality conditions among non-edges. For which values of d is the Lovasz-Saks-Schrijver ideal of G radical,
prime or complete intersection? These are the questions we will discuss in the talk. Furthermore we will show how this study isrelated to the investigation of special sections of determinantal varieties. For example we show that in characteristic 0 the idealof 3-minors of a generic matrix with some entries replaced by 0 is always reduced and that the analogous statement for 4minors does not hold in general.
The talk is based on a joint paper with Volkmar Welker (Marburg).

Dale Cutkosky (University of Missouri)
Mixed Multiplicities of Filtrations

We develop a theory of mixed multiplicities for possibly non Noetherian filtrations of m-primary ideals.
We show that many of the classical theorems for mixed multiplicities of m-primary ideals are true in this more general situation.
An example which we will discuss is of filtrations associated to m-valuations, which are particularly nice in this theory, eventhough they are generally not Noetherian. This talk will include joint work with Parangama Sarkar, Hema Srinivasan and JugalVerma.

Hai Long Dao (University of Kansas)
Loaded Modules

Over a commutative Noetherian local ring R, we will study the notion of loaded modules. These objects can be shown to include
Ulrich modules and integrally closed ideals, and still hold many well-known desirable properties.

David Eisenbud (MSRI & University of California Berkeley)Residual Intersections for Chasles to Ulrich

I'll briefly sketch the history of residual intersection theory from the 3264 conics of Chasles through the work of Fulton, ArtinNagata and Huneke; and then concentrate on the work of Bernd Ulrich and his co-authors on the question of when a residualintersection is Cohen-Macaulay (and refinements).

Louiza Fouli (New Mexico State University)
The Core of Ideals

The core of an ideal in a Noetherian local ring with infinite residue field is defined as the intersection of all its reductions andencodes information about all possible reductions. In the first part of the talk we will give a brief survey of this topic highlighting
the contributions of Professor Bernd Ulrich. In the second part of the talk we will discuss some new results concerning the coreof monomial ideals. This is joint work with Jonathan Monta\~{n}o, Claudia Polini, and Bernd Ulrich.

Mel Hochster (University of Michigan)
Faithfulness of Top Local Cohomology

Let 𝑅𝑅 be a local domain and 𝐼𝐼 a proper ideal of 𝑅𝑅. Let d be the cohomological dimension of 𝐼𝐼, so that 𝐻𝐻^𝑑𝑑_𝐼𝐼(𝑅𝑅) is not 0 but all
higher local cohomology modules vanish. It is an open question, raised in the thesis of L. Lynch, whether 𝐻𝐻^𝑑𝑑_𝐼𝐼(𝑅𝑅) must be afaithful 𝑅𝑅-module. The talk will discuss the status of this question and report on joint work Jack Jeffries which proves
faithfulness when the arithmetic rank of 𝐼𝐼 is 𝑑𝑑 provided that 𝑅𝑅 has characteristic 𝑝𝑝 > 0. So far as we know, the question remains
open when 𝐼𝐼 has arithmetic rank 𝑑𝑑 and 𝑅𝑅 contains the rational numbers.

Craig Huneke (University of Virginia)

The Mathematical Work of Bernd Ulrich
This talk will survey some of the many highlights of Bernd's mathematics, ranging through Berger's conjecture, linkage, residualintersections, Gorenstein rings, Rees algebras, integral closures, the core of ideals, and equisingularity.

Srikanth Iyengar (University of Utah)
Modules with no higher self-Tors

The following question was raised, implicitly, by Liana Şega:If a finitely generated module M over a local ring R satisfies Tor_i(M,M)=0 for i>>0, is then M of finite projective dimension? I
will report on progress on this question, made in collaboration with Avramov, Nasseh, and Sather-Wagstaff.

Steve Kleiman (MIT)

Macaulay Duality over Any Base
Traditionally, Macaulay Duality furnishes a useful canonical bijective correspondence between Artinian quotients of apolynomial ring over a base field and modules of linear functionals on forms. In joint work in progress with Jan Kleppe of Oslo,this duality is generalized over an arbitrary Noetherian base ring, thus providing a suitable framework for studying a family ofArtinian quotients by investigating its dual family.

Andrew Kustin (University of South Carolina)

Resolutions which are Differential Graded Algebras
Let P be a commutative Noetherian ring and F be a self-dual acyclic complex of finitely generated free P-modules. Assume that F
has length four and F_0 has rank one. We prove that F can be given the structure of a Differential Graded Algebra with Divided
Powers; furthermore, the multiplication on F exhibits Poincare Duality. This result is already known if P is a local Gorenstein ringand F is a minimal resolution. The purpose of the present project is to remove the unnecessary hypotheses that P is local, P isGorenstein, and F is minimal.

Linquan Ma (Purdue University)
Asymptotic Lech’s inequality

In 1960 Lech proved a simple inequality that relates the Hilbert-Samuel multiplicity and the colength of an m-primary ideal in a
Noetherian local ring. However, this inequality is almost never sharp. In this talk, we discuss optimal versions of Lech's inequality
for sufficiently deep ideals. We prove them using Hilbert-Kunz theory in characteristic p>0, and we conjecture that they hold inarbitrary characteristic. This is joint work with Craig Huneke, Pham Hung Quy, and Ilya Smirnov.

Claudia Miller (Syracuse University)

Betti numbers of the Frobenius powers of the maximal ideal over generic hypersurfaces in 3 variables.
We discuss the conjectured curious behavior of the Betti numbers of the Frobenius powers of the maximal ideal in
hypersurfaces $R=k[x,y,z]/(f)$, where $k$ is a field of positive characteristic $p$, as described by Kustin and Ulrich. We provethat if $f$ is chosen suitably generically (with $k$ uncountable) and has degree of opposite parity to $p$ then high enough
Frobenius powers of the maximal ideal have identical graded Betti numbers up to explicit shifts. We also find the Hilbert-Kunz
function and some structural features of the resolution and its associated matrix factorization. The proof is based on a technicalresult on the growth of the socles in the quotients, which in turn is based on our constructions of free resolutions of relatively
compressed Artinian graded algebras. This is joint work with Hamid Rahmati and Rebecca R.G.

Viet Trung Ngo (Vietnam Academy of Science and Technology)
Depth functions of powers of homogeneous ideals

A classical result of Brodmann says that the numerical non-negative function depth R/I^t, where I is a homogeneous ideal in apolynomial ring R, is always convergent, i.e. it is constant for t large enough. In 2005, Herzog and Hibi posed the conjecture thatthe function depth R/I^t can be any numerical non-negative convergent function. This talk will will give an affirmative answer
to this conjecture and shows a similar result for the symbolic powers I^(t) of I, namely that for any numerical positive function
f(t) which is periodic for t large enough, there is a homogeneous ideal I in a polynomial ring R such that depth R/I^(t) = f(t) for t >0. It is still an open question whether the function depth R/I^(t) is always periodic for t large enough.

Irena Peeva (Cornell University)
Regularity of prime ideals

This talk will discuss some natural questions about the regularity of prime ideals. We give counterexamples to the EisenbudGoto Regularity Conjecture using Rees algebras, and also counterexamples that do not rely on the Mayr-Meyer construction.
Furthermore, examples of prime ideals for which the difference between the maximal degree of a minimal generator and themaximal degree of a minimal first syzygy can be made arbitrarily large are provided. Using a recent result of Ananyan-Hochster
we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone. This is joint work with
Giulio Caviglia, Marc Chardin, Jason McCullough, and Matteo Varbaro.

Marilina Rossi (University of Genoa, Italy)

The Castelnuovo-Mumford regularity of the “blow-up algebras”
The Castelnuovo-Mumford regularities of the Rees algebra, of the associated graded ring and of the fiber ring of an ideal arestrongly related. We present a survey on this topic and we show how the stability of the Ratliff-Rush filtration and the invarianceof reduction number of the ideal give interesting consequences on the regularity of these graded algebras. In the case ofhomogeneous ideals generated in the same degree these methods recently led to a counterexample to a conjecture stated by
D. Eisenbud and B. Ulrich.

Anurag Singh (University of Utah)

Differential operators on some invariant rings
Work of Levasseur and Stafford describes the rings of differential operators on various classical invariant rings of characteristic
zero; in each of these cases, the differential operators form a simple ring. Towards an attack on the simplicity of rings ofdifferential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked
if reduction modulo p works for differential operators in this context. In joint work with Jack Jeffries, we establish that this is
not the case for certain SL-invariants.

Karen Smith (University of Michigan)
Cluster Algebras in Prime Characteristic

Cluster Algebras are commutative algebras introduced by Sergey Fomin and Andrei Zelevinsky in order to capture some of theunifying ideas occurring throughout combinatorics, representation theory and total positivity. Defined by a simple combinatorial
process, the commutative algebraic properties of cluster algebras have not yet been deeply studied. In this talk, I will introduce
cluster algebras, show that upper cluster algebras are always Frobenius split, and that locally acyclic cluster algebras are always
F-regular. This is joint work with Angelica Benito, Greg Muller and Jenna Rajchgot.

Shunsuke Tagaki (University of Tokyo, Japan)
Vanishing theorems on globally F-regular varieties
Globally F-regular varieties are a special type of Frobenius split varieties and closely related to log Fano varieties. Toric andSchubert varieties (in positive characteristic) are important examples of globally F-regular varieties. I will discuss vanishingtheorems, and in particular Kollar's injectivity theorem for semi-ample line bundles, on globally F-regular varieties. This talk isbased on joint work with Yoshinori Gongyo.

Kevin Tucker (University of Illinois - Chicago)
p_g Vector Bundles on Normal Surfaces
It is well-known that complete (or integrally closed) ideals on a rational surface singularities satisfy a number of special
properties -- for example, a product of complete ideals is complete. While similar statements fail for general normal surfaces,
they hold for a special class of complete ideals introduced by Okuma-Watanabe-Yoshida called p_g ideals. In this talk, I will
discuss a higher-rank generalization of these concepts, called p_g vector bundles and some of their basic properties. This is joint
work with Lawrence Ein.

Kei-ichi Watanabe (Nihon University, Japan)

Normal reduction numbers of integrally closed ideals in a 2-dimensional cone singularities.
This is a joint work in progress with Tomoyuki Okuma and Ken-ichi Yoshida.
Let (𝐴𝐴, 𝑚𝑚, 𝑘𝑘) be an excellent normal 2-dimensional local ring and 𝐼𝐼 be an integrally closed 𝑚𝑚-primary ideal. We assume 𝑘𝑘 is
algebraically closed and 𝑘𝑘 ⊂ 𝐴𝐴.
For any resolution of singularity 𝑓𝑓: 𝑋𝑋 → 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(𝐴𝐴), we define 𝑝𝑝𝑔𝑔 (𝐴𝐴) = 𝑑𝑑𝑑𝑑𝑑𝑑𝑘𝑘 𝐻𝐻1 (𝑋𝑋, 𝒪𝒪𝑋𝑋 ). 𝐴𝐴 is a rational singularity if 𝑝𝑝𝑔𝑔 (𝐴𝐴) = 0.
For a minimal reduction 𝑄𝑄 of 𝐼𝐼, we calculate the numbers;
𝑁𝑁 , ∀𝑁𝑁 ≥ 𝑛𝑛�
nr(𝐼𝐼) = min{𝑛𝑛 | ������
𝐼𝐼𝑛𝑛+1 = 𝑄𝑄𝐼𝐼�𝑛𝑛 }, 𝑟𝑟̅ (𝐼𝐼) = min�𝑛𝑛 � ������𝐼𝐼𝑁𝑁+1 = 𝑄𝑄𝐼𝐼���
andnr(𝐴𝐴) = max{nr(𝐼𝐼)}, 𝑟𝑟̅(𝐴𝐴) = max {𝑟𝑟̅ (𝐼𝐼)} (𝐼𝐼 ⊂ 𝐴𝐴, 𝑚𝑚 − primary integrally closed ideals)
Example. 𝑟𝑟̅(𝐴𝐴) = 1 if and only 𝐴𝐴 is a rational singularity and if 𝐴𝐴 is an elliptic singularity, then 𝑟𝑟̅(𝐴𝐴) = 2. In general, we can show𝑟𝑟̅(𝐴𝐴) ≤ 𝑝𝑝𝑔𝑔 (𝐴𝐴) + 1.
We discuss 𝐴𝐴̅ in case of cone singularities (we say 𝐴𝐴 is a cone singularity if the exceptional set of the minimal resolution of 𝐴𝐴 is asingle smooth irreducible curve) using a vanishing theorem of Röhr. In particular, we will show;
Theorem. If 𝐴𝐴 is a hypersurface singularity of dimension 2 defined by a homogeneous polynomial of degree 𝑑𝑑 ≥ 4, then 𝑟𝑟̅(𝐴𝐴) =nr(𝑚𝑚) = 𝑑𝑑 − 1.
Also, we will show an example of 𝐴𝐴 and 𝐼𝐼 such that nr(𝐼𝐼) = 1 and 𝑟𝑟̅ (𝐼𝐼) = 𝑝𝑝𝑔𝑔 (𝐴𝐴) + 1 for every 𝑝𝑝𝑔𝑔 (𝐴𝐴) ≥ 2.